enhanced trailer backingup a trailer. for an autonomous vehicle with a trailer, backing with a...
TRANSCRIPT
ENHANCED TRAILER BACKING
By
AMIT DAVID JAYAKARAN
A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2004
Copyright 2004
by
Amit David Jayakaran
This document is dedicated to my parents.
iv
ACKNOWLEDGMENTS
I would like to express my gratitude to Dr. Carl Crane III, who as my advisor came
up with the idea for this thesis and who guided me for the past year and a half. I would
also like to thank Dr. David Dooner for his immense help in getting my work started and
putting me on the right track. I would also like to thank Dr. Robert Bicker who helped me
through the last part of my work. I would like to thank Dr. John Kenneth Schueller for
being on my committee.
I also owe a lot of gratitude towards Shannon Ridgeway and Dr. Carol Chesney
who helped me at various points during my work.
On a more personal note I would like to thank my parents and brother for their love
and support. I would also like to thank my friends at the University of Florida who have
directly and indirectly contributed to my work here.
v
TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................................................................................. iv
LIST OF TABLES............................................................................................................ vii
LIST OF FIGURES ......................................................................................................... viii
ABSTRACT...................................................................................................................... xii
CHAPTERS
1 INTRODUCTION ............................................................................................................1
2 REVIEW OF LITERATURE ...........................................................................................4
Overview.......................................................................................................................4 Definitions ....................................................................................................................5
Jack Knifing...........................................................................................................5 Instant Center of Rotation .....................................................................................5 Radius of Turn.......................................................................................................6 Curvature of Path...................................................................................................6 Nonholonomic Constraints....................................................................................6 Small Time Controllable .......................................................................................7 Off-Hooking ..........................................................................................................7 Center of Rotation .................................................................................................8
3 APPROACH .....................................................................................................................9
Assumptions .................................................................................................................9 Approach.......................................................................................................................9 Stable Values of Theta................................................................................................13
Analytical Solution for the Polygon ....................................................................13 The Simulink Model............................................................................................18
Velocity Analysis........................................................................................................22 Stabilizing the Trailer .................................................................................................27
4 DESIGN DESCRIPTION...............................................................................................30
Mechanisms to Align Trailer ......................................................................................30
vi
Method I ..............................................................................................................30 Method II .............................................................................................................31
Implementation ...........................................................................................................36 Simulink ..............................................................................................................36 Physical Model ....................................................................................................38
Data acquisition-Potentiometers ..................................................................38 The Servo .....................................................................................................42
The Prototype..............................................................................................................43 Rotary to Linear Conversion ...............................................................................43 Mounting of Potentiometers ................................................................................46
Trailer angle potentiometer ..........................................................................46 Steering angle potentiometer........................................................................46 x value feedback potentiometer....................................................................46
Approximation of Sine Using Taylor Series Expansion.............................................47 Implementation ...........................................................................................................54
5 RESULTS AND CONCLUSIONS ................................................................................55
APPENDIX SOFTWARE MODEL OF THE TRAILER PROBLEM .............................62
LIST OF REFERENCES...................................................................................................72
BIOGRAPHICAL SKETCH .............................................................................................74
vii
LIST OF TABLES
Table page 4-1: Sample Values for the Setup ......................................................................................36
4-2: High-torque Ball-bearing Servo Motor from GWS....................................................42
4-3: Taylor Substitution .....................................................................................................48
viii
LIST OF FIGURES
Figure page 1-1: Jackknifing ...................................................................................................................1
2-1: Instant Center of Rotation ............................................................................................6
2-2: Center of rotation..........................................................................................................8
3-1: Combined Instant Center............................................................................................10
3-2: Combined Instant Center............................................................................................10
3-3: Simplified system and the solution polygon ..............................................................11
3-4: Solution 1 for the Angle θ ..........................................................................................12
3-5: Solution 2 for the Angle θ ..........................................................................................12
3-6: Calculation of the Distance of the Instant Center from the Center of the Rear Axle.14
3-7: Trailer and Vehicle Polygon......................................................................................15
3-8: Triangle to Help Simplify the Solution ......................................................................17
3-10: Reading Constants - I ...............................................................................................20
3-11: Reading Constants - II ..............................................................................................20
3-12: Calculation of the Instant Center - I .........................................................................20
3-13: Calculation of the Instant Center - II ........................................................................20
3-14 Solution for First Theta Value ...................................................................................21
3-15 Solution for Second Theta Value...............................................................................21
3-16: Plot of Turning Radius .............................................................................................22
3-17: The Setup for the Slider Analysis.............................................................................24
3-18: Closer Look at the Velocity Setup at the End of the Trailer ....................................25
ix
3-19: Velocities at the Trailer ............................................................................................26
3-20: Slider Setup. .............................................................................................................29
4-1: Imparted Slider Motion ..............................................................................................31
4-2: Algorithm for Method I ..............................................................................................32
4-3: Algorithm for Method II.............................................................................................33
4-4: Graphs of Input Parameters. .......................................................................................34
4-5: Offset vs. Time ...........................................................................................................35
4-6: Trailer Angle vs. Time ...............................................................................................35
4-7: Path and Trailer Orientation .......................................................................................37
4-8: Traditional Resistor Setup ..........................................................................................38
4-9: Modified Resistor Setup .............................................................................................39
4-10: High-Torque Ball-Bearing Servo Motor from GWS................................................43
4-11: The Prototype Setup – I............................................................................................44
4-12: The Prototype Setup – II...........................................................................................44
4-13: The Working of the Slider........................................................................................45
4-14: Mounting of the Pulley on the Servo........................................................................45
4-15: Trailer Angle Potentiometer .....................................................................................47
4-16: Steering Angle Potentiometer...................................................................................49
4-17: 1st Order Series Truncation.......................................................................................50
4-18: 3rd Order Series Truncation ......................................................................................50
4-19: 5th Order Series Truncation ......................................................................................51
4-20: 7th Order Series Truncation ......................................................................................51
4-21: Error Plots for Sine Approximation .........................................................................52
4-22: 2nd Order Series Truncation......................................................................................52
4-23: 4th Order Series Truncation ......................................................................................53
x
4-24: 6th Order Series Truncation ......................................................................................53
4-25: Error Plots for Cosine Approximation .....................................................................54
5-1: Steering Angle Acquired at Each Time Step..............................................................55
5-2: Slider Displacement, x, at Each Time Step ................................................................56
5-3: Relative Angle θ at Each Time Step ..........................................................................56
5-4: Value of y(cm) at Each Time Step .............................................................................57
5-5: At A and B..................................................................................................................58
5-6: At D ............................................................................................................................58
5-7: Between D and E ........................................................................................................58
5-8: At E.............................................................................................................................59
5-9: Between F and G ........................................................................................................59
5-10: Between G and H......................................................................................................60
5-11: At H and I .................................................................................................................60
5-12: At J, K and L ............................................................................................................60
A-1: Base Model ................................................................................................................62
A-2: Setting System Constants – Masked..........................................................................63
A-3: Setting System Constants – Under the Mask.............................................................63
A-4: Setting System Constants – Dialog Box....................................................................64
A-5: Vehicle Speed Input...................................................................................................64
A-6: Vehicle Speed Input Sent to Global From-Block......................................................65
A-7: Convert Vehicle Speed from a Ramp to a Smooth Curve.........................................65
A-8: Steering Input ............................................................................................................65
A-9: Steering Input Sent to Global From-Block................................................................66
A-10: Central Control of Model ........................................................................................66
A-11: A Delay Function to Make the Model Realistic ......................................................67
xi
A-12: Current Theta Processing.........................................................................................67
A-13: Required Theta Processing ......................................................................................68
A-14: Calculation of New Required Theta – I...................................................................68
A-15: Calculation of New Required Theta – II .................................................................68
A-16: Calculation of i (Distance of the Instant Center from the Rear Wheel) ..................69
A-17: Calculation of a, the Turning Radius.......................................................................69
A-18: Calculation of Required Offset................................................................................69
A-19: Calculation of the Correction in Offset Required - I ...............................................70
A-20: Calculation of the Correction in the Offset Required – II .......................................70
A-21: Animation of Model ................................................................................................70
xii
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science
ENHANCED TRAILER BACKING
By
Amit David Jayakaran
May 2004
Chair: Carl D. Crane III Major Department: Mechanical and Aerospace Engineering
In many instances, an autonomous vehicle has a trailer attached to it. The trailer
readily follows the vehicle’s path of motion while it moves forward. However, when the
vehicle reverses, for example to correct its path or to get out of a tight corner, the trailer
has a tendency to jackknife.
This thesis deals with several approaches to solve this problem, none of which
requires any modification to the vehicle steering control itself. The approach dealt with
in detail involves actually moving the point that couples the trailer to the vehicle.
Depending on the instant center of rotation of the vehicle with respect to the ground,
changes are made in the position of the coupling point to modify the instant center of the
trailer with respect to ground in such a manner as to maintain stability during backing.
The results obtained are verified in both a computer simulation and a physical model. The
results were found to be excellent with the vehicle being able to back up with the trailer
through any distance at moderate speeds.
1
CHAPTER 1 INTRODUCTION
When a vehicle backs up with a trailer attached the trailer never seems to follow the
path it is expected to. Drivers with a lot of driving experience can even be often confused
by an uncooperative trailer while backing. It takes a lot of experience to be able to back
up a trailer. For an autonomous vehicle with a trailer, backing with a trailer is a really big
challenge as it places additional constraints on the system’s path of travel.
Jackknifing is defined as the condition when a trailer bends or folds up on the
vehicle pulling it, similar to a jackknife shown in Figure 1-1. As far as backing up with a
trailer most work done so far involves increasing the number of constraints on the vehicle
in order to prevent jackknifing which thereby reduces the probability of the vehicle
following a defined path closely. Basically most algorithms will end up causing the
vehicle to make extra or at least wider turns. This may in some cases be unacceptable.
Figure 1-1: Jackknifing
2
The following thesis deals with different approaches to prevent jackknifing without
having to place any extra constraints on the vehicle. It aides the autonomous vehicle to
back up with a trailer as if it were not there at all. This basically means all that is needed
to do is to place the necessary sensors on any vehicle and it will be able to back up
without having to change anything on the vehicle. The design of this system enables it to
be transferred to a regular vehicle attached to a trailer, thus aiding a human in backing a
trailer.
Jackknifing is chiefly caused due to incorrect backing of a vehicle with a trailer.
Jackknifing can also be caused during forward motion when the vehicle brakes suddenly
and the trailer continues to move and since it can’t move forward it swings sideways and
jackknives. However this thesis will deal with jackknifing caused due to backing only.
Such a system is of particular interest for an autonomous vehicle. Many autonomous
vehicles carry their surveillance equipment in a trailer and so the path that the trailer
follows is very important. Besides this sometimes the vehicle may find itself in a tight
corner or some situation in which it may need to back up a particular distance, causing it
to jackknife.
The vehicle is nonholonomic because of its rolling constraints. The configuration
of the vehicle is given by two position coordinates and an angle and there are only two
inputs, linear and angular velocity. Thus the system has two degrees of freedom. For
every trailer that is added, the configuration of the vehicle depends on one more
constraint that is the angle between the trailer and what it is attached to.
3
Thesis Organization
The following thesis is divided into five chapters.
The second chapter, Review of Literature, deals with the literature review and past
work done in this area.
The third chapter, Approach, involves a look at the different approaches that can be
used to solve the given problem. It lists the assumptions made during the entire analysis
and derives the equations required for the analysis and implementation of the model.
The fourth chapter, Design Description gives a description of the computer model
and then the physical model that was built based on chapter three.
The fifth chapter, Results and Conclusions, describes the results that are obtained
and also the future work that is possible in this area.
4
CHAPTER 2 REVIEW OF LITERATURE
Overview
In the past much work has been done on stabilizing trailers during their forward
motion for different speeds, especially high speeds. For example Tesar and Matthew [1]
discuss a four bar mechanism that helps eliminate swaying at high speeds.
Most work in path planning of a vehicle with a trailer has been done by
introduction of additional constraints on the system [2-7]. A vehicle already has
nonholonomic constraints and the introduction of additional constraints further restricts
the path the vehicle is able to take. The work done on Hilare and its trailer [5] outlines
several equations for backing up with a trailer with and without off-hooking. These
approaches though successful, reduce the efficiency of path planning because of the
introduction of additional constraints.
The backing up of a vehicle trailer system can be compared to that of an inverted
pendulum. In an inverted pendulum the system is stable only in only one position [8-9].
In order to stabilize the inverted pendulum it is moved horizontally to ensure that that it
always remains vertical. With a trailer system too this is partly true. However the stable
point keeps changing depending on the wheel angle, because for different wheel angles
the trailer chooses another stable angle.
5
To further facilitate the understanding of some of the terms used in this thesis some
terms are defined below.
Definitions
Jack Knifing
Jackknifing is defined as the condition when a trailer bends or folds up on the
vehicle pulling it, similar to a jackknife.
Instant Center of Rotation
Instant Center of Rotation is defined as the imaginary point about which the vehicle
appears to be turning with respect to ground. The instant center of rotation is at infinity
when the vehicle is moving straight. It is located at the center of the minimum turning
radius of the vehicle when the vehicle is making the sharpest possible turn it can make.
Ackerman steering is typically used in steered wheeled vehicles so that the instant
center is located at the point at which the perpendiculars from all the wheels meet. As
long as these lines meet at only one point the system is stable. Vehicle manufacturers
make sure these lines meet at the designated point when turning. Figure 1 shows the
instant center of rotation for a wheeled vehicle with Ackerman steering.
6
Figure 2-1: Instant Center of Rotation
Radius of Turn
Radius of Turn is the distance between the instant center of rotation and the center
of the rear axle of a front wheel steered vehicle.
Curvature of Path
The inverse of the radius of turn is called the curvature. Only in the case that the
vehicle is a two wheel system is it possible for it to follow a curvature of infinity.
Nonholonomic Constraints
Nonholonomic constraints arise from the fact that the wheels of a four wheeled
vehicle can only roll or spin on the ground but allow for no slide sideways [2]. For
instance a vehicle has two controls – linear and angular velocities. However this vehicle
moves in a three dimensional configuration space. As a result, all paths on a
configuration space are not feasible. To add to this, when there is a trailer it further
reduces the possibilities. For example, if a vehicle has to back up straight it can do so
without a problem if there is no trailer. But when a trailer is attached it will have to steer
7
gently right and left continuously in order to make sure the trailer doesn’t jackknife. This
becomes tougher when negotiating a turn as the vehicle has to now make a turn in the
opposite direction first to align the trailer in the direction the trailer has to go and then
only start negotiating the turn. This can be more clearly seen by observing a person
backing up with a trailer. This thesis outlines a method by which some constraints on the
system associated with the trailer may be eliminated [3]. Nonholonomic equations are
characterized by constraint equations involving the time derivatives of the system
configuration variables. These equations are non integrable; they typically arise when the
system has less control than the configuration variables. The rolling contact of the
wheels with the surface it moves on is what essentially gives rise to nonholonomic
motion [10-11]
Small Time Controllable
Small-time controllability means that the set of configurations reachable after any
given time always contains a neighborhood of the starting configuration. As a
consequence any collision free path can be approximated by a sequence of collision free
feasible configurations [5].
Off-Hooking
Off-hooking is when the trailer is hooked at a distance from the rear axle of the
vehicle.
8
Center of Rotation
The center of rotation is defined as the point of rotation of the trailer with respect to
the vehicle. It is usually the point at which the trailer is attached to the vehicle.
Figure 2-2: Center of rotation
9
CHAPTER 3 APPROACH
Assumptions
The following assumptions about the system are presented:
• It is assumed that the contact between the ground and each every wheel is pure rolling contact
• All the systems are planar.
Approach
When a trailer is attached to the system, there is one more factor that is now
contributing to the instant center of rotation. When this vehicle moves forward the trailer
follows the vehicle and will negotiate turns without any problems. As shown later the
trailer naturally tries to align itself with the instant center of rotation of the vehicle by
rotating about its center of rotation at the hinge. The trailer being aligned with the instant
center of rotation of the vehicle simply means that the line through the trailer axle passes
through the instant center of rotation of the vehicle. However, when backing up the
vehicle with the trailer, the system is as unstable as trying to balance an inverted broom
on one’s hand. There are only two positions at which the trailer will be stable and out of
these two positions, as shown later only one really is feasible. What makes this problem
worse is that this position of stability changes as the steering angle of the vehicle
10
changes. Thus if it can be somehow ensured that the line through the wheel axle of the
trailer passes through the instant center of rotation of the vehicle, the system will be
stable. In other words, if the instant center of the trailer with respect to ground is always
coincident with the instant center of the vehicle with respect to ground, then the vehicle
and trailer can travel in reverse in a stable manner. This condition is shown in Figure 3-1
and 3-2.
Figure 3-1: Combined Instant Center
Figure 3-2: Combined Instant Center
11
This thesis outlines alternative methods for coupling between the vehicle and the
trailer to ensure that the instant center of rotation is always the same for the vehicle as
well as the vehicle trailer system. It also describes in detail a generic system that can be
extended to any vehicle-trailer system in the market.
The trailer angle corresponding to different steering angles can be calculated by
many methods. Below are outlines of some of the methods that have been used in this
thesis to cross verify results.
The above system is simplified into a geometric system and is represented in
Figure 3-3. The values for ‘d’, the distance of the trailer hitch from the rear axle of the
vehicle, ‘l’, the distance from the trailer hitch to the trailer wheel axle, and ‘a’ the
distance between the instant center point and the center of the vehicle’s rear axle, are
assumed to be known.
Figure 3-3: Simplified system and the solution polygon
12
It is found that there are two possible solutions for the angle θ, the angle between
the line along the trailer and the line along the vehicle’s rear axle, for this system. These
solutions are show in the Figures 3-4 and 3-5 below.
Figure 3-4: Solution 1 for the Angle θ
Figure 3-5: Solution 2 for the Angle θ
Obviously the first solution is the only one of interest because the second solution
would represent an existing case of jackknifing. The interesting point about the second
solution is that in this condition the trailer is apparently being pulled when the vehicle
travels in reverse. As a result it will naturally align itself to achieve the angle to coincide
with the instant center of rotation. A method to find the value for θ for a given value of
the steering angle φ (shown in Figure 3-6) that will keep the trailer stable while backing,
13
i.e. the value of θ that will cause the line along the trailer axle pass through the instant
center of rotation of the vehicle, will be presented in the next section. After deriving the
stable position of the trailer different methods to keep the trailer stable at those positions
will be discussed.
Stable Values of θ
Analytical Solution for the Polygon
The analytical method involves converting the system into a polygon and finding
the various solutions that satisfy it. The polygon is shown in Figure 3-7. The aim is to
find a solution to the angle θ in relation to ‘a’, the distance of the instant center from the
center of the rear axle. The distance of the instant center is calculated from the wheel
angle (φ), the width of the vehicle (w ) and the distance between the rear and front axles
(L1).
The distance of the instant center ( i ) from the rear wheel is given by
1 tan2
i L π = × −Φ
(3.1)
Now the term of interest in is ‘a’, the distance between the center of the axle and
the instant center which is given by
2wa i= + (3.2)
Once this is determined, the parameters for the polygon are calculated. The terms
x, y, and z are defined as shown in Figure 3-7.
14
Figure 3-6: Calculation of the Distance of the Instant Center from the Center of the Rear Axle
Figure 3-7 shows the trailer and the vehicle polygon. The distances x and y can be
written in terms of the trailer rotation angle θ as
cosdxθ
= (3.3)
tany d θ= (3.4)
From the outer triangle of Figure 3-7 which has sides of length z, a+y, and l+x it is
apparent that
cos2
l xa y
π θ + − = + . (3.5)
15
Figure 3-7: Trailer and Vehicle Polygon
From equations (3.3), (3.4) and (3.5) the following expression can be written
( )( )
coscos
2 tan
dl
a dθπ θθ
+ − = +
(3.6)
This equation relates the trailer angle,θ , to the instant center distance, a . The left
side of (3.6) will equal sinθ . Making this substitution and multiplying both sides by
( tan )a d θ+ gives
sin cos ( tan )cosda d lθ θ θθ
+ = + (3.7)
Multiplying throughout by cosθ gives
sin cos ( tan ) cosa d l dθ θ θ θ+ = + (3.8)
Substituting sintancos
θθθ
= and regrouping gives
16
2(sin cos ) (sin 1) cos 0a d lθ θ θ θ+ − − = (3.9)
Substituting 2 2(sin 1) cosθ θ− = − and dividing throughout by cosθ gives
sin cos 0a d lθ θ− − = (3.10)
Two values of θ will satisfy this equation for each given value of a . One of these
values will be physically realizable as shown in Figures 3-4 and 3-5.
There are two methods of solving this equation [12]. Rewriting the above equation
as
cos sin 0A B Dθ θ+ + = (3.11)
where
, ,A d B a D l= − = = −
Trigonometric Solution
This technique begins by dividing (3.11) by 2 2A B+ to yield
2 2 2 2 2 2
cos sin 0A B DA B A B A B
θ θ+ + =+ + +
(3.12)
Using the right-angled triangle shown in Figure 3-8 it is possible to substitute
2 2
sinBA B
γ=+
(3.13)
2 2
cosAA B
γ=+
(3.14)
Because the sine and cosine of γ are expressed in terms of all known quantities, a
unique value for the angle γ can be obtained. Substituting (3.13) and (3.14) into (3.12)
gives
17
2 2
cos cos sin sin 0DA B
γ θ γ θ+ + =+
(3.15)
Figure 3-8: Triangle to Help Simplify the Solution
Using the trigonometric identity
cos( ) cos cos sin sinα β α β α β− = + (3.16)
Putting (3.16) into (3.15) and regrouping gives
2 2
cos( ) DA B
θ γ −− =
+ (3.17)
Thus there are two values for the quantity ( )θ γ− because the cosine yields two
values in the range 0 2θ π≤ ≤ . But sinceγ is unique two value of θ can now be
obtained.
18
The Simulink Model
A Simulink model was built based on the results discussed above to plot the desired
values of the trailer angle with respect to the steering angle of the vehicle. This is a
component of the main program sued to make a computer model of the complete system
described in more detail in the Appendix.
Figure 3-10 illustrates the base model. It takes the vehicle constants namely,
vehicle width, distance of trailer hitch (off-hooking), length of trailer, and distance of rear
and front axles (wheel base) as the input. The steering angle is simulated with a ramp
that runs from 0 to 70 degrees. The system is symmetric and so the same conclusions can
be deduced from the -70 to 0 degree range. Figure 3-11 and Figure 3-12 are for reading
constants from the user and making them globally available. Figure 3-13 and 3-14
calculate constants that are required during the simulation. The results of these blocks are
fed into a Matlab function that outputs the θ values that are stable, i.e., the values of θ
that will cause the line through the trailer axle to pass through the vehicle’s instant center
point. The results are then plotted to view the response of the steering angle on trailer
position.
The various parameters were set as
• Distance between the front and rear axles = 1.5 m • Width of the vehicle = 1 m • Distance of hinge from the rear axle = 0.3 m • Length of the trailer = 1 m
The output is in the form of the two possible solutions of theta. The results were
viewed on a virtual scope and are shown in Figures 3-14 and 3-15. Besides this the
turning radius is also shown in Figure 3-16.
19
Figu
re 3
-9: B
ase
Mod
el fo
r θ C
alcu
latio
n
20
Figure 3-10: Reading Constants - I
Figure 3-11: Reading Constants - II
Figure 3-12: Calculation of the Instant Center - I
Figure 3-13: Calculation of the Instant Center - II
21
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
80
90
Steering Angle (deg)
Trai
ler A
ngle
(deg
)
Figure 3-14 Solution for First Theta Value
0 10 20 30 40 50 60 70120
130
140
150
160
170
180
Steering Angle (deg)
Trai
ler A
ngle
(deg
)
Figure 3-15 Solution for Second Theta Value
22
0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
80
90
100
Steering Angle (deg)
Turn
ing
Rad
ius
(m)
Figure 3-16: Plot of Turning Radius
Velocity Analysis
So far all calculations are done not considering the vehicle motion at all. Thus in
order to verify these results a velocity analysis is performed. The velocity analysis will
result in a relation between the trailer angle, change in the trailer angle and the velocity of
the vehicle. Now in this relation the change in trailer angle is set to zero as this is what is
desired at steady state. If the relation is similar to those obtained earlier then the above
results are verified.
Let the vehicle be moving at a velocity of v and the average wheel turning angle be
φ . Both wheels are usually at almost the same angle and both contribute to the radius of
turn. In order to consider both their contributions the average is taken.
23
The following information is assumed to be given:
• the wheel base of the vehicle is 1L • the amount of off-hooking is rl • The length of the trailer tl
The following notation is introduced:
• Orientation of the vehicle with the x axis τ • Angle between the vehicle and the trailer θ • Turning radius a
The setup is shown in Figure 3-17.
The angular velocity of the vehicle about its instant center of rotation is given by,
va
ω = (3.18)
Now,
1tan La
φ = (3.19)
Rewriting,
1
tanLaφ
= (3.20)
From (3.20) and (3.18)
1
tanvLφω = (3.21)
The omega is now translated to the center of the rear axle [13-18]. This leaves an
angular velocity of ω about the center of the rear axle and a velocity v as shown in Figure
3-18. Thus the linear velocity of the center point of the rear axle is a counter
clockwiseω and a velocity v in the direction of the front wheels.
24
Figure 3-17: The Setup for the Slider Analysis
Now it is known that
G T G V V Tθ θ θ= + (3.22)
Differentiating (3.22)
G T G V V Tθ θ θ= + (3.23)
25
Figure 3-18: Closer Look at the Velocity Setup at the End of the Trailer
It is required to find V Tθ ,
V T G V G Tθ θ θ= − (3.24)
It is known that
G Vθ ω= − (3.25)
The negative sign is put there becauseω is counter clockwise.
The velocity at the hinge has two components. One is from the velocityv of the
vehicle and the other caused due to the angular velocity, ω , of the vehicle. The
velocityv is at an angle ofτ to the x-axis. The velocity caused due to ω ( rl ω ) is
perpendicular to the off hooking.
26
Figure 3-19: Velocities at the Trailer
The component of the velocity that causes the trailer to rotate about the center point
of its axle is
cos cos2rl v πω θ θ + −
. (3.26)
Thus the angular velocity of the trailer is
cos sinrG T
t t
l vl lωθ θ θ= + . (3.27)
Substituting (3.27) and (3.25) into (3.24) gives
cos sinrV T
t t
l vl lωθ ω θ θ= − − − . (3.28)
27
This equation is the control equation. By finding values to push V Tθ to zero when
in steady state, the trailer can be made stable during backing.
The correct value of θ is obtained when 0V Tθ = . Thus (3.28) reduces to
cos sin 0r
t t
l vl lωω θ θ− − − = (3.29)
Dividing throughout by v and substituting equation (3.18) into (3.29) gives
1 1cos sin 0r
t t
la al l
θ θ− − − = (3.30)
cos sin 0t rl l aθ θ+ + = (3.31)
It is seen that this equation is equivalent to equation (3.48), thus verifying the
results. Another interesting fact is that the desired trailer angle is independent of the
vehicle speed when the system is at a stable condition. Thus most calculations for stable
conditions can be done without knowledge of the velocity of the vehicle.
Stabilizing the Trailer
Stabilizing the trailer involves trying to change θ to the desired angle. This boils
down to a control problem where orientation of the trailer is being controlled with just
one control parameter. There are several approaches of doing this.
One approach that is relatively simple to implement is to place a motor at the hinge
point and force the trailer to take the angle that is calculated from above. This approach is
of course feasible only in cases where the trailer weight is very low and where the vehicle
doesn’t steer unless it is in motion. There will be very high levels of slippage when the
28
trailer is being forced to an angle but once it achieves the correct desired angle, there will
be little to no slippage.
Another approach involves placing a non linear gear train between the steering and
the trailer. This way whenever the vehicle is steering, the trailer gets appropriately
steered too. This is similar to the above approach except that the link is now mechanical
and not electrical. Besides the disadvantages stated above this approach also has the
added disadvantage of the increased steering force required.
Slider
In all the above approaches the trailer is pushed to a particular position which is
stable. Thus in these cases the ideal trailer angle is known. Another method would be to
slide the hinge of the trailer along the back of the vehicle to such an extent so as to rotate
the trailer to the desired trailer angle. This can be done by means of placing a slider on
the back of the trailer as seen in Figure 3-17. As the trailer angle changes the slider
moves to compensate it.
Now from reproducing equation (3.28)
cos sinrV T
t t
l vl lωθ ω θ θ= − − − (3.32)
From the above equation it is seen that the only term that will be affected when the
slider moves is the v term corresponding to the velocity of the vehicle at the hinge. v was
earlier calculated as
v rω= (3.33)
Now it will be given by
29
( )v r x ω= + (3.34)
Figure 3-20: Slider Setup.
Thus using equation (3.34) in (3.32) the required results are obtained. However
usually for reasonable turning angles x will be very small compared to r and so it can be
ignored. So for all practical purposes the results obtained in equation (3.28) are
sufficient.
A method of implementation of this control system is discussed in Chapter 4.
30
CHAPTER 4 DESIGN DESCRIPTION
Mechanisms to Align Trailer
The objective, as stated in the previous chapter is to obtain stable backing of the
vehicle and trailer by causing the angle θ to be at such a value that the line along the
trailer axle will pass through the instant center point of the vehicle during the backing
maneuver. Two approaches will be presented that force the trailer to remain properly
aligned as the vehicle is driven in reverse. A physical model to test the approaches is
then described. Results of tests with the physical model are presented in Chapter 5.
Equation (3.31) is being repeated here as (4.1) cos sin 0t rl l aθ θ+ + = (4.1)
Method I
The simplest most straight forward method of ensuring that the system is stable is
to always ensure that the trailer angle satisfies the above equation. In this case all that
needs to be done is to ensure that the Equation (4.1) is satisfied and if it not the slider is
moved such that it is. This is shown in Figure 4-1.
31
Figure 4-1: Imparted Slider Motion
The algorithm is illustrated in Figure 4-2.
Method II
A new variable ‘y’ is introduced that corresponds to the position of the center of the
trailer axle, measured parallel to the slider. Ideally this point (y=0) would always be in
line with the line through the center of axles of the vehicle. Thus the slider is moved to a
position such that the trailer takes the angle just calculated. Now for some reason if the y
value changes to some other value x is moved in such a way to position the trailer relative
to the vehicle such that y equals zero again.
32
This method uses jackknifing to its own benefit. When the value of y isn’t close to
zero, the slider moves to a position such that the trailer begins to jackknife in the opposite
direction and when y is at the desired value the slider moves back to its original position.
This is illustrated in Figure 4-3.
Figure 4-2: Algorithm for Method I
Besides this depending on the steering angle the initial value of x is calculated
knowing the value of θ corresponding to every steering angle. Due to the lower
processing power of the microprocessor the calculation of the desired trailer angles were
33
done offline and some sample values are tabulated below. Interpolation is then used to
calculate intermediate values. From this the x values were obtained.
Figure 4-3: Algorithm for Method II
34
0 50 100 150 200 250-50
-40
-30
-20
-10
0
10
20
30
40
50
Time (sec)
Ste
erin
g A
ngle
(deg
)
Figure A
0 50 100 150 200 2500
1
2
3
4
5
6
Time (sec)
Spe
ed (m
/s)
Figure B
Figure 4-4: Graphs of Input Parameters.
A) Steering Angle vs. Time B) Speed vs. Time
35
0 50 100 150 200 250-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (sec)
Offs
et (m
)
Figure 4-5: Offset vs. Time
0 50 100 150 200 250-50
-40
-30
-20
-10
0
10
20
30
Time (sec)
Trai
ler A
ngle
(deg
)
Figure 4-6: Trailer Angle vs. Time
36
φ θ
-12 13.2649
-9 9.8876
-6 6.5631
-3 3.2730
0 0
3 -3.2730
6 -6.5631
9 -9.8876
12 -13.2649
Table 4-1: Sample Values for the Setup
Implementation
Simulink
The Simulink model involves recreating the system response of a vehicle-trailer
model with introduction of the slider at the hinge of the trailer. This system implements
the first algorithm described above. The second algorithm is implemented physically and
hence isn’t discussed here. The system takes the steering angle and the speed of the
vehicle as input. Speed is taken as an input because the system response will vary
depending on the speed of the vehicle because unlike the speed of the vehicle the slider
can only move within limitations. Thus a vehicle backing up at a very high speed may not
37
be easy to control. It also takes the current offset and the trailer angle as an input. From
this it calculates the required offset and moves the slider to that position in order to
compensate any change in the steering angle or change in trailer angle caused due to
maybe uneven terrain. It also has noise in the steering angle input which helps recreate a
more realistic model. The Simulink model is shown in the Appendix in more detail.
Below are the results of the simulation. The steering angle (Figure 4-4a) and the
vehicle speed (Figure 4-4b) are the inputs to the system and accordingly the offset
(Figure 4-5) is changed and thus the trailer’s angle (Figure 4-6). Figure 4-7 shows the
path that the vehicle follows and its orientation at various points on that curve.
Figure 4-7: Path and Trailer Orientation
38
Physical Model
The physical model is built on a small remote controlled car. The vehicle features
digital steering which allowed the steering angle of the car to be changed to any desired
angle. The car was modified to attach a trailer and a slider.
Figure 4-8: Traditional Resistor Setup
Data acquisition-Potentiometers
The potentiometers used were of the rotary type with outputs at the two ends and
one on the wiper. The range of the potentiometer was approximately 180˚. This implied
that the potentiometer’s resistance between one of the end terminals and the wiper would
39
change from almost zero to almost the resistance between the end terminals for a rotation
of about 180˚ and vice versa for the other end terminal.
From the above setup (Figure 4-8) it is seen that the voltage can be varied across
the resistor between 0 and 5 volts. However for the experimental setup a range of 45˚ to
135˚ is only required. Besides this in order to have a wider range of values it was also
required to have the voltage at 45˚ relative to ground to be 0 and the voltage at 135˚
relative to ground to be 2.5.
In order to do this two unknown resistances 1R and 2R are added in series as shown
in Figure 4-9.
Figure 4-9: Modified Resistor Setup
1 2total potR R R R= + + (4.2)
40
( )1 180
poth l
a ltotal
aRR V V
V VR
+ −
= + (4.3)
( )1 180
poth l
b ltotal
bRR V V
V VR
+ −
= + (4.4)
From these equations 1R and 2R is calculated as below
( )
( )1 180a l l b pot
b a
V b aV bV V a RR
V V− − + +
= −−
(4.5)
( )
2
180 180180( )
h h a b b a pot
b a
aV bV V b V a V V RR
V V− + − + −
= −−
(4.6)
Now in for this case the requirements were
4.79
45135
4.765.0502.5
pot
l
h
a
b
R k
abV VV VV VV V
= Ω
= °= °= −===
(4.7)
Thus
1
2
3.2761.245
R kR k= Ω= Ω
(4.8)
Also in some cases the angle of the range may not be known but resistances at the
end points are known. In that case let aR and bR be the resistances
( )( )1 a h la l
total
R R V VV V
R+ −
= + (4.9)
41
( )( )1 b h lb l
total
R R V VV V
R+ −
= + (4.10)
From these equations 1R and 2R are calculated as below
( )( )1
b a a l a b b l
b a
V R R V V R R VR
V V− + + −
= −−
(4.11)
( )
2 ( )a h b h b a b pot a b a pot
b a
R V R V V R V R V R V RR
V V− − + + −
= −−
(4.12)
Now in for this case the requirements were
4.84
2.0752.7064.76
5.0502.5
pot
a
b
l
h
a
b
R k
R kR kV VV VV VV V
= Ω
= Ω= Ω= −
===
(4.13)
Thus
1
2
0.8961.490
R kR k= − Ω= − Ω
(4.14)
Resistances can’t be less than zero and so the range of the potentiometer was
increased. aR was decrease and bR was increases by 500kΩ .
Thus
1.5753.206
a
b
R kR k
= Ω= Ω
(4.15)
This yielded
1
2
1.4710.029
R kR k= Ω= Ω
(4.16)
42
The Servo
The servo is an actuation device that will rotate to any commanded position. The
servo used takes three inputs. Two are for power out of which one is the common ground
and the other is +5 volts. The third input is a PWM whose duty cycle controls the
position of the servo to any desired angle within its range. The duty cycle of the PWM
can be varied from between 10% to 20% with a time period of 20ms to get the servo to
rotate between the zero position and the 180 degree position.
Dimensions (mm)
1.56 x 0.79 x 1.56 in.
39.5 x 20.0 x 39.6 mm
Weight 46 Grams / 1.62 oz
Ball Bearings Yes, 2BB
Metal Gears No
Torque (4.8V) 69 oz.in.
Transit Time (4.8V)
0.21 sec./60°
Torque (6.0V) 86 oz.in.
Transit Time (6.0V)
0.17 sec./60°
Table 4-2: High-torque Ball-bearing Servo Motor from GWS
43
The Prototype
Rotary to Linear Conversion
The rotary motion of the servo is converted to a linear motion by means of a pulley
– slider system. The setup is shown in Figure 4-11 and 4-12.
When the pulley rotates the slider moves changing the trailer angle as it does this.
In Figure 4-13 the slider is moved to the left forcing the trailer to change its angle.
Figure 4-13 shows the linking of the pulley with the slider along the back of the
vehicle. Figure 4-14 shows the mounting of the pulley on the servo attached to the body
of the vehicle.
Figure 4-10: High-Torque Ball-Bearing Servo Motor from GWS
44
Figure 4-11: The Prototype Setup – I
Figure 4-12: The Prototype Setup – II
45
Figure 4-13: The Working of the Slider
Figure 4-14: Mounting of the Pulley on the Servo
46
Mounting of Potentiometers
Trailer angle potentiometer
The potentiometer to read the trailer angle is mounted inside a servo casing and
attached to the slider. This is seen in Figure 4-15.
Steering angle potentiometer
The potentiometer to read the steering angle is mounted below the hinge for one of
the front wheels of the car. Although this gives the reading for only one wheel and not the
average, it was found that this value was sufficient to get the prototype to work. The
mounting is shown in Figure 4-16.
x value feedback potentiometer
The servo though very accurate in getting to a position that is requested like any
mechanical system takes a while to get where it is commanded. However the servo used
did not have any feedback to let the microcontroller know what its current position was.
Thus a modification had to be made so that is would output its current position. This was
done by tapping the potentiometer built into the servo and calculating the voltage drop
across it.
47
Figure 4-15: Trailer Angle Potentiometer
Of course the best way to get the value of X would be to actually introduce a linear
transducer on the slider. However, this wasn’t cost effective and would make the slider
very large and also be an additional load on the servo, decreasing its response time.
Reading the position off the servo gave the value of x within a reasonable level of
accuracy and so this value was converted and used.
Approximation of Sine Using Taylor Series Expansion
Given a function whose first n derivatives can be found, the Taylor polynomial
( )nP x of degree n for the function ( )f x is defined about the value c as [19]
( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )2 3
... ...2! 3! !
nn
n
f c x c f c x c f c x cP x f c f c x c
n′′ ′′′− − −
′= + − + + + + (4.17)
where ( ) ( )nf c denotes the nth derivative of the function f evaluated at x c= .
48
It is required to find the Taylor Series expansion for
( ) sinf x x= (4.18)
The derivatives of sin x are first found and then evaluated at 0c = (A Taylor Series
at 0c = is also known as a MacLaurin series)
n ( ) ( )nf x ( ) ( )nf c
0 sin x 0
1 cos x 1
2 sin x− 0
3 cos x− -1
4 sin x 0
5 cos x 1
6 sin x− 0
7 cos x− -1
8 sin x 0
9 cos x 1
10 sin x− 0
Table 4-3: Taylor Substitution
( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )2 3
sin ... ...2! 3! !
nnf c x c f c x c f c x cx f c f c x c
n′′ ′′′− − −
′= + − + + + + (4.19)
( ) ( )( ) ( )( ) ( )( )2 3sin 0 0 cos 0 0sin sin 0 cos 0 0 ...
2! 3!x x
x x− − − −
= + − + + + (4.20)
3 5 7 9
sin ...3! 5! 7! 9!x x x xx x= − + − + (4.21)
49
Similarly for ( ) cosf x x= (4.22)
2 4 6 8
cos 1 ...2! 4! 6! 8!x x x xx = − + − + + (4.23)
The assumption of ( )sin x x= for small values of x can be verified by truncating
the Taylor Series to just the first order term. The results are seen in the plot Figure 4-17.
Figure 4-16: Steering Angle Potentiometer
Since the processor used wasn’t very powerful a check was done to see how long
an approximation of the Taylor series was required to get sufficiently accurate values for
sine and cosine. Below are plots for truncated series of sine and cosine based on the order
of the assumption. Figure 4-17, 4-18, 4-19, and 4-20 show the sine value approximated in
red to the first order, third order, fifth order and seventh order respectively. The actual
sine plot is also there in blue to show a comparison. Figure 4-21 shows the error between
the actual value and the value obtained for all the series truncations. A strange feature is
50
noticed with the 5th order approximation of sine which appears to be more accurate than
the 7th order approximation.
Figure 4-17: 1st Order Series Truncation - sin x x=
Figure 4-18: 3rd Order Series Truncation - 3
sin3!xx x= −
51
Figure 4-19: 5th Order Series Truncation - 3 5
sin3! 5!x xx x= − +
Figure 4-20: 7th Order Series Truncation - 3 5 7
sin3! 5! 7!x x xx x= − + −
52
Figure 4-21: Error Plots for Sine Approximation
It is seen from the above graphs that the 5th order sine gives sufficient accuracy in
the range of -90˚ to 90˚. Thus
3 5
sin3! 5!x xx x== − + (4.24)
Figure 4-22: 2nd Order Series Truncation - 2
cos 12!xx = −
53
A similar analysis is done for the cosine plots in Figures 4-22 to 4-25.
Figure 4-23: 4th Order Series Truncation - 2 4
cos 12! 4!x xx = − +
Figure 4-24: 6th Order Series Truncation - 2 4 6
cos 12! 4! 6!x x xx = − + −
54
Figure 4-25: Error Plots for Cosine Approximation
It is seen from the above graphs that the 6th order cosine gives sufficient accuracy
in the range of -90˚ to 90˚. Thus
2 4 6
cos 12! 4! 6!x x xx = − + − (4.25)
Implementation
The model was build and programmed as per the second algorithm described
above. The results of the tests run on the model are described in the next chapter.
55
CHAPTER 5 RESULTS AND CONCLUSIONS
A control loop was written to change the position of the slider based on the second
method of implementation as described in the previous chapter.
To best illustrate the working of the algorithm the vehicle was backed up and the
data was acquired into a computer and plotted. The values read in were the current values
of steering angle, x (slider position) and trailer angle. Pictures were also taken at some
positions to illustrate the data. y is calculated for each point and plotted too. For ease of
understanding the time scale is divided in sections labeled A-L.
Figure 5-1: Steering Angle Acquired at Each Time Step
56
Figure 5-2: Slider Displacement, x, at Each Time Step
Figure 5-3: Relative Angle θ at Each Time Step
57
Figure 5-4: Value of y(cm) at Each Time Step
From Figure 5-1 it is seen that the steering angle remains practically stable at about
3˚. The change that occurred is due to the flexibility in the steering system. The vehicle is
considered to be stable when the value of y lies between -1.5 to 1.5 cm. The stable
condition is defined as the state at which the trailer is least likely to jackknife. For a
steering angle of 3 degrees the appropriate required trailer angle is about -3.3°.
Until time step C the vehicle isn’t moving. The trailer angle till this point remains
stable at about -8°.
As the vehicle backs up the trailer angle decreases to beyond -10°, thus increasing
the value of y to below -1.5 cm. Thus the slider changes the value of x to around -6cm
between C and D.
58
Figure 5-5: At A and B
Figure 5-6: At D
Figure 5-7: Between D and E
59
This changes θ to about 13° and as the vehicle backs up y begins to change until it
reaches a position that is within the tolerable limits. At this point, between D and E the
slider returns to its normal position.
Figure 5-8: At E
However the slider does not react very fast. As a result the θ value again goes out
of sync and begins to reduce. To compensate the slider now moves to +8.5cm between E
and F, remaining at that position till the value of y reaches within the allowable range.
Figure 5-9: Between F and G
60
Figure 5-10: Between G and H
Figure 5-11: At H and I
Figure 5-12: At J, K and L
61
The slider then moves back to the normal position and over a period of time the
trailer regains its correct θ at about G. Similar corrections occur until time step J and the
system thereafter remains stable.
Thus it is seen the system remains stable so that backing up doesn’t cause the trailer
to jackknife. The trailer is able to back up with reasonable efficiency at a reasonable
speed.
An added benefit discovered was that while the vehicle is moving forward and
negotiating a turn the slider can move outward causing the trailer to make a wider turn.
This means that the vehicle will not have to make wide turns when turning sharp. Thus
when a trailer is attached to the vehicle the vehicle can now drive forward also to an
extent without worrying about the trailer.
The mechanism discussed in this thesis is simple enough to implement on almost
any vehicle, be it autonomous or regular. The addition enables any driver without any
particular skills in trailer backing to be able to back a vehicle up through any path.
As far as future work goes, methods can be introduced to increase the efficiency of
the backing system so that the slider doesn’t have to travel as much as it currently does.
This would enable the trailer to move more efficiently and follow the vehicle’s path more
closely. Also, as on now the vehicle is the one that is being modified. Work can be done
such that the attachment is made on the trailer and so it can be used on any vehicle by the
simple attachment of a steering sensor.
62
APPENDIX SOFTWARE MODEL OF THE TRAILER PROBLEM
The following is the software model of the vehicle-trailer system. The basic blocks
shown in Figure A-1 are the system process block which outputs to an animation block.
The system process block takes its input from the constants definition block and also a
predefined steering angle and velocity values described later. Theta is the angle of the
trailer, the offset is the distance of the slider from its central position and ‘a’ is the turning
radius of the vehicle.
Figure A-1: Base Model
Figure A-2, A-3 and A-4 are the constant setting blocks. The vehicle width,
distance of trailer hitch (off-hooking), length of trailer and distance of rear and front axels
(wheel base) can be set here so that they are accessible by the model at any time.
63
Figure A-2: Setting System Constants – Masked
Figure A-3: Setting System Constants – Under the Mask
64
Figure A-4: Setting System Constants – Dialog Box
Figures A-5, A-6 and A-7 show how the vehicle speed is read from a data file and
sent to the system. There is a delay block and a saturation block to make the system
response to velocity commands slower in response.
Figure A-5: Vehicle Speed Input
65
Figure A-6: Vehicle Speed Input Sent to Global From-Block
Figure A-7: Convert Vehicle Speed from a Ramp to a Smooth Curve
Figures A-8 and A-9 show how the steering angle of the vehicle is read from a data
file and made available to the entire system.
Figure A-8: Steering Input
66
Figure A-9: Steering Input Sent to Global From-Block
Figure A-10 is the central control and simulation of the model. The first part of the
system takes in various values as its input and the second part simulates its effect on a
real vehicle-trailer system. Figure A-11 shows a simple delay block that delays the
response of the slider so that it doesn’t move to the commanded position instantaneously.
This makes the simulation more realistic.
Figure A-10: Central Control of Model
67
Figure A-11: A Delay Function to Make the Model Realistic
In a real system the value of Theta would be simply read off a sensor but in the
case of a software model this has to be simulated as shown in Figure A-12. Besides this
the required Theta to make the system stable is also required and this is calculated as
shown in Figures A-12, A-13, A-14 and A-15.
Figure A-12: Current Theta Processing
68
Figure A-13: Required Theta Processing
Figure A-14: Calculation of New Required Theta – I
Figure A-15: Calculation of New Required Theta – II
For the system to run other values have to be calculated as illustrated in the figures
below. Figure A-16 shows how the distance of the instant center from the rear wheel is
calculated. Figure A-17 shows how the turning radius of the vehicle is calculated.
69
Figure A-16: Calculation of i (Distance of the Instant Center from the Rear Wheel)
Figure A-17: Calculation of a, the Turning Radius
After the required Theta is calculated, the offset required to change the trailer angle
is calculated as shown in Figures A-18, A-19 and A-20.
Figure A-18: Calculation of Required Offset
70
Figure A-19: Calculation of the Correction in Offset Required - I
Figure A-20: Calculation of the Correction in the Offset Required – II
Figure A-21: Animation of Model
71
These values are fed back into the system and the whole cycle starts again for the
next time step.
Figure A-21 shows the values that are fed into the animation system for the model.
Figure A-22 helps output all values to graphs for debugging as well as for plotting
outputs.
Figure A-22: Debug Box for Troubleshooting and viewing outputs
72
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15. McGill, David J., Engineering Mechanics: an Introduction to Dynamics, 2nd ed., Boston, Mass.: PWS -Kent Pub. Co., 1989.
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17. Shabana, Ahmed A., Dynamics of Multibody Systems, New York: Wiley, 1989.
18. Hibbeler, R. C., Engineering Mechanics-Dynamics, 4th ed., New York: Macmillan Pub. Co.; London: Collier Macmillan Publishers, 1986.
19. J. A. Sellers, “Section 1. Why Taylor Series,” Available at URL: http://www.krellinst.org/UCES/archive/resources/taylor_series/node1.shtml, Accessed in 08/2003.
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BIOGRAPHICAL SKETCH
Amit David Jayakaran was born on May 1st, 1979, in India. He received this
Bachelor of Engineering in mechanical engineering from the College of Engineering,
Anna University, in India in April 2001.
He joined the University of Florida for his master’s in mechanical engineering. He
has been a research assistant working on enhanced trailer backing with the Mechanical
and Aerospace Engineering Department at the University of Florida. His research
interests include controls, robotics and industrial automation.